Small subspaces of \(L_p\). (English) Zbl 1234.46017

The investigation of the subspaces of the classical Banach space \(L_p=L_p[0,1]\) has been in the centre of functional analytic interest since the days of Banach. It has long been known that the subspace structure of \(L_p\) is different in the ranges \(p<2\) and \(p>2\); for example \(L_r\) is isometric to a subspace of \(L_p\) if \(1\leq p\leq r\leq 2\), but this is not so if \(2<p<\infty\) and \(r\notin\{2,p\}\). This paper makes a deep contribution to understanding the subspaces of \(L_p\) for \(p>2\). Throughout this review \(p\) will always denote a real number greater than \(2\).
By now classical results due to M. Kadets and A. Pełczyński on the one hand and W.B. Johnson and E. Odell on the other hand imply, for a subspace \(X\subset L_p\), that if \(X\) is not hilbertian, then \(X\) contains a copy of \(\ell_p\), and if \(X\) does not contain a copy of \(\ell_2\), then \(X\) embeds into \(\ell_p\). The authors point out a combination of these results as the starting point for their studies: If \(X\subset L_p\) neither embeds into \(\ell_p\) nor into \(\ell_2\), then \(X\) contains a copy of \(\ell_p\oplus \ell_2\), which isolates \(\ell_p\oplus \ell_2\) as another “small” subspace of \(L_p\).
The main result of the paper takes this idea one step further: If \(X\subset L_p\) does not embed into \(\ell_p\oplus \ell_2\), then \(X\) contains a copy of \(\ell_p(\ell_2)\). Indeed, a more precise version is proved in Theorem B: If \(X\subset L_p\) does not embed into \(\ell_p\oplus \ell_2\), then for every \(\varepsilon>0\) there is a subspace \(Y_\varepsilon\subset X\) that is \((1+\varepsilon)\)-isometric to \(\ell_p(\ell_2)\) and complemented in \(L_p\) by a projection of norm \(\leq (1+\varepsilon)\gamma_p\), where \(\gamma_p\) denotes the \(L_p\)-norm of a standard Gaussian variable, and this estimate is optimal.
The proof of this result is very involved and difficult, and it takes the better part of this well written paper to complete it. The first step is an intrinsic characterisation of those subspaces of \(L_p\) that embed into \(\ell_p\oplus \ell_2\) (Theorem A); a simplified version of this result reads as follows: \(X\subset L_p\) embeds into \(\ell_p\oplus \ell_2\) if and only if for every normalised weakly null tree in \(X\) there exists a branch \((x_i)\) such that for some \(K\geq1\) and all finite sets of scalars one has \[ \frac1K \Bigl\|\sum a_i x_i \Bigr\|_{L_p} \leq \|(a_i)\|_{\ell_p} + \Bigl\|\sum a_i x_i \Bigr\|_{L_2} \leq K \Bigl\|\sum a_i x_i \Bigr\|_{L_p}. \] The proof of this result, given in Section 3, depends on the techniques developed by Odell, Schlumprecht and their coauthors in the last decade.
In Section 4 a dichotomy of Kadets-Pełczyński type for subspaces \(X\) of \(L_p\) is proved. Its first half leads to the conclusion that \(X\) embeds into \(\ell_p\oplus \ell_2\); for this, inequalities due to Rosenthal and Burkholder play an important role. Section 5 is devoted to the proof that in the alternative case of the dichotomy \(\ell_p(\ell_2)\) \((1+\varepsilon)\)-embeds into \(X\). Section 6 details the argument that there is even a well-complemented \((1+\varepsilon)\)-copy of \(\ell_p(\ell_2)\); this proof relies on Aldous’s theory of random masures and the Krivine-Maurey theory of stable Banach spaces. A blend of both approaches that are commonly regarded as two faces of the same medal is needed here. Incidentally, the authors also record an easier proof due to G. Schechtman to obtain a \((1+\varepsilon)\)-copy of \(\ell_p(\ell_2)\) that is complemented by some norm.
The final two sections contain miscellaneous material: a new proof of the theorem due to W. B. Johnson and E. Odell [“Subspaces and quotients of \(\ell_p\oplus \ell_2\) and \(X_p\),” Acta Math.147, 117–147 (1981; Zbl 0484.46020)] saying that a subspace of \(L_p\) that is isomorphic to a quotient of \(\ell_p\oplus \ell_2\) in fact embeds into \(\ell_p\oplus \ell_2\), a discussion of the impossibility of a uniform (let alone almost isometric) embedding in the context of Theorem A, and remarks about \({\mathcal L}_p\)-subspaces of \(\ell_p\oplus \ell_2\) and Rosenthal’s space \(X_p\).


46B25 Classical Banach spaces in the general theory
46B03 Isomorphic theory (including renorming) of Banach spaces
46B06 Asymptotic theory of Banach spaces


Zbl 0484.46020
Full Text: DOI arXiv


[1] D. J. Aldous, ”Subspaces of \(L^1\), via random measures,” Trans. Amer. Math. Soc., vol. 267, iss. 2, pp. 445-463, 1981. · Zbl 0474.46007 · doi:10.2307/1998664
[2] D. E. Alspach, ”Tensor products and independent sums of \(\mathcalL_p\)-spaces, \(1&lt;p&lt;\infty\),” Mem. Amer. Math. Soc., vol. 138, iss. 660, p. viii, 1999. · Zbl 0926.46012
[3] D. Alspach and E. Odell, ”\(L_p\) spaces,” in Handbook of the Geometry of Banach Spaces, Vol. I, Amsterdam: North-Holland, 2001, pp. 123-159. · Zbl 1008.46013 · doi:10.1016/S1874-5849(01)80005-X
[4] J. Bourgain, H. P. Rosenthal, and G. Schechtman, ”An ordinal \(L^p\)-index for Banach spaces, with application to complemented subspaces of \(L^p\),” Ann. of Math., vol. 114, iss. 2, pp. 193-228, 1981. · Zbl 0496.46010 · doi:10.2307/1971293
[5] D. L. Burkholder, ”Distribution function inequalities for martingales,” Ann. Probability, vol. 1, pp. 19-42, 1973. · Zbl 0301.60035 · doi:10.1214/aop/1176997023
[6] D. L. Burkholder, B. J. Davis, and R. F. Gundy, ”Integral inequalities for convex functions of operators on martingales,” in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, 1972, pp. 223-240. · Zbl 0253.60056
[7] I. S. Èdel\('\)vsteuin and P. Wojtaszczyk, ”On projections and unconditional bases in direct sums of Banach spaces,” Studia Math., vol. 56, iss. 3, pp. 263-276, 1976. · Zbl 0362.46017
[8] D. J. H. Garling, ”Stable Banach spaces, random measures and Orlicz function spaces,” in Probability Measures on Groups, New York: Springer-Verlag, 1982, vol. 928, pp. 121-175. · Zbl 0485.46012
[9] Y. Gordon, D. R. Lewis, and J. R. Retherford, ”Banach ideals of operators with applications,” J. Functional Analysis, vol. 14, pp. 85-129, 1973. · Zbl 0272.47024 · doi:10.1016/0022-1236(73)90031-1
[10] S. Guerre-Delabrière, Classical Sequences in Banach Spaces, New York: Marcel Dekker, 1992, vol. 166. · Zbl 0756.46007
[11] <a href=’http://links.jstor.org/sici?sici=0091-1798(199010)18:42.0.CO;2-8&origin=MSN’ title=’Go to document’> P. Hitczenko, ”Best constants in martingale version of Rosenthal’s inequality,” Ann. Probab., vol. 18, iss. 4, pp. 1656-1668, 1990. · Zbl 0725.60018 · doi:10.1214/aop/1176990639
[12] W. B. Johnson, ”On quotients of \(L_p\) which are quotients of \(l_p\),” Compositio Math., vol. 34, iss. 1, pp. 69-89, 1977. · Zbl 0375.46023
[13] W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, ”Symmetric structures in Banach spaces,” Mem. Amer. Math. Soc., vol. 19, iss. 217, p. v, 1979. · Zbl 0421.46023
[14] W. B. Johnson and E. Odell, ”Subspaces of \(L_p\) which embed into \(l_p\),” Compositio Math., vol. 28, pp. 37-49, 1974. · Zbl 0282.46020
[15] W. B. Johnson and E. Odell, ”Subspaces and quotients of \(l_p\oplus l_2\) and \(X_p\),” Acta Math., vol. 147, iss. 1-2, pp. 117-147, 1981. · Zbl 0484.46020 · doi:10.1007/BF02392872
[16] W. B. Johnson and J. Lindenstrauss, ”Basic concepts in the geometry of Banach spaces,” in Handbook of the Geometry of Banach Spaces, Vol. I, Amsterdam: North-Holland, 2001, pp. 1-84. · Zbl 1011.46009 · doi:10.1016/S1874-5849(01)80003-6
[17] W. B. Johnson, H. P. Rosenthal, and M. Zippin, ”On bases, finite dimensional decompositions and weaker structures in Banach spaces,” Israel J. Math., vol. 9, pp. 488-506, 1971. · Zbl 0217.16103 · doi:10.1007/BF02771464
[18] W. B. Johnson and M. Zippin, ”On subspaces of quotients of \((\sum G_n)_{lp}\) and \((\sum G_n)_{c_0}\),” in Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces, 1972, pp. 311-316 (1973). · Zbl 0252.46025 · doi:10.1007/BF02762805
[19] M. I. Kadec and A. Pełczyński, ”Bases, lacunary sequences and complemented subspaces in the spaces \(L_p\),” Studia Math., vol. 21, pp. 161-176, 1961/1962. · Zbl 0102.32202
[20] N. J. Kalton and D. Werner, ”Property \((M)\), \(M\)-ideals, and almost isometric structure of Banach spaces,” J. Reine Angew. Math., vol. 461, pp. 137-178, 1995. · Zbl 0823.46018 · doi:10.1515/crll.1995.461.137
[21] H. Knaust, E. Odell, and T. Schlumprecht, ”On asymptotic structure, the Szlenk index and UKK properties in Banach spaces,” Positivity, vol. 3, iss. 2, pp. 173-199, 1999. · Zbl 0937.46006 · doi:10.1023/A:1009786603119
[22] A. Koldobsky and H. König, ”Aspects of the isometric theory of Banach spaces,” in Handbook of the Geometry of Banach Spaces, Vol. I, Amsterdam: North-Holland, 2001, pp. 899-939. · Zbl 1005.46005 · doi:10.1016/S1874-5849(01)80023-1
[23] J. -L. Krivine and B. Maurey, ”Espaces de Banach stables,” Israel J. Math., vol. 39, iss. 4, pp. 273-295, 1981. · Zbl 0504.46013 · doi:10.1007/BF02761674
[24] J. Lindenstrauss and A. Pełczyński, ”Absolutely summing operators in \(L_p\)-spaces and their applications,” Studia Math., vol. 29, pp. 275-326, 1968. · Zbl 0183.40501
[25] J. Lindenstrauss and H. P. Rosenthal, ”The \(\mathcalL_p\) spaces,” Israel J. Math., vol. 7, pp. 325-349, 1969. · Zbl 0205.12602 · doi:10.1007/BF02788865
[26] D. A. Martin, ”Borel determinacy,” Ann. of Math., vol. 102, iss. 2, pp. 363-371, 1975. · Zbl 0336.02049 · doi:10.2307/1971035
[27] B. Maurey, V. D. Milman, and N. Tomczak-Jaegermann, ”Asymptotic infinite-dimensional theory of Banach spaces,” in Geometric Aspects of Functional Analysis, Basel: Birkhäuser, 1995, vol. 77, pp. 149-175. · Zbl 0872.46013
[28] E. Odell, ”On complemented subspaces of \((\sum l_2)_{lp}\),” Israel J. Math., vol. 23, iss. 3-4, pp. 353-367, 1976. · Zbl 0333.46005 · doi:10.1007/BF02761814
[29] E. Odell and T. Schlumprecht, ”Trees and branches in Banach spaces,” Trans. Amer. Math. Soc., vol. 354, iss. 10, pp. 4085-4108, 2002. · Zbl 1023.46014 · doi:10.1090/S0002-9947-02-02984-7
[30] E. W. Odell and T. Schlumprecht, ”Embedding into Banach spaces with finite dimensional decompositions,” RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., vol. 100, iss. 1-2, pp. 295-323, 2006. · Zbl 1118.46018
[31] E. Odell, T. Schlumprecht, and A. Zsák, ”On the structure of asymptotic \(l_ p\) spaces,” Q. J. Math., vol. 59, iss. 1, pp. 85-122, 2008. · Zbl 1156.46013 · doi:10.1093/qmath/ham026
[32] A. Pełczyński, ”Projections in certain Banach spaces,” Studia Math., vol. 19, pp. 209-228, 1960. · Zbl 0104.08503
[33] A. Pełczyński and H. P. Rosenthal, ”Localization techniques in \(L^p\) spaces,” Studia Math., vol. 52, pp. 263-289, 1974/75. · Zbl 0297.46023
[34] C. Rosendal, ”Infinite asymptotic games,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 59, iss. 4, pp. 1359-1384, 2009. · Zbl 1187.46006 · doi:10.5802/aif.2467
[35] H. P. Rosenthal, ”On the subspaces of \(L^p\) \((p&gt;2)\) spanned by sequences of independent random variables,” Israel J. Math., vol. 8, pp. 273-303, 1970. · Zbl 0213.19303 · doi:10.1007/BF02771562
[36] G. Schechtman, ”Examples of \(\mathcalL_p\) spaces \((1&lt;p\not=2&lt;\infty )\),” Israel J. Math., vol. 22, iss. 2, pp. 138-147, 1975. · Zbl 0316.46018 · doi:10.1007/BF02760162
[37] G. Schechtman.
[38] M. Zippin, ”Banach spaces with separable duals,” Trans. Amer. Math. Soc., vol. 310, iss. 1, pp. 371-379, 1988. · Zbl 0706.46015 · doi:10.2307/2001128
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